Cabbage: A Differential Growth Framework for Open Surfaces

TL;DR

The paper addresses modeling buckling in 3D open surfaces using differential growth. It introduces Cabbage, a pipeline that grows meshes, applies smoothing, enforces collision constraints, and converts results to CAD-ready NURBS. Compared to state-of-the-art methods, Cabbage achieves higher morphology expressiveness, better mesh quality, and robust self-intersection avoidance, while remaining open-source and accessible. This enables digital fabrication, education, and data generation for geometry processing and shape analysis.

Abstract

We propose Cabbage, a differential growth framework to model buckling behavior in 3D open surfaces found in nature-like the curling of flower petals. Cabbage creates high-quality triangular meshes free of self-intersection. Cabbage-Shell is driven by edge subdivision which differentially increases discretization resolution. Shell forces expands the surface, generating buckling over time. Feature-aware smoothing and remeshing ensures mesh quality. Corrective collision effectively prevents self-collision even in tight spaces. We additionally provide Cabbage-Collision, and approximate alternative, followed by CAD-ready surface generation. Cabbage is the first open-source effort with this calibre and robustness, outperforming SOTA methods in its morphological expressiveness, mesh quality, and stably generates large, complex patterns over hundreds of simulation steps. It is a source not only of computational modeling, digital fabrication, education, but also high-quality, annotated data for geometry processing and shape analysis.

Figures (2)

• Figure 1: The proposed Cabbage pipeline is composed of $T$ iterative applications of the Differential Growth block to an initial mesh $M_0 = (\mathcal{V}, \mathcal{E}, \mathcal{F})$ followed by a CAD model conversion step. In the former, mesh connectivity is optimized with edge flips, collapses, and ear removal. Growth factors $g(v) \forall v\in\mathcal{V}$ are computed as a function Eq \ref{['eq:growth-factor']} of geodesic distance to select vertices of the mesh, and they drive differential edge subdivision. The mesh surface is deformed by bending and stretching forces and optional external fields. Global and local feature-aware smoothing ensures mesh fairness. Vertex-vertex collisions prevent self-intersections. After desired morphology develops, the triangular mesh $M_\Delta$ is quadrangulated, offset, and parametrized by a polysurface composed of individual NURBS patches $\{S_i(u, v)|i \in I \}$
• Figure 2: Left to right: Morphology development at simulation steps $0, 100, 200, 300, 400, 500$. Top to bottom: initial mesh of momeomorphic types Disk $D^2$, Annulus (Ring, $S^1 \times I$), Möbius Band, and Punctured Torus ($T^2 \backslash D^2$). Boundary edges are highlighted.